Travelling in Schengen area while waiting for permit of stay renewal in Italy
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It is possible and you need to apply for a residence permit called: Permesso di soggiorno provvisorio.
Here is a list of documents you need to submit to the police station:
The expired residence card
The receipt of the post office for the renewal request
Original passport
4 Photos
16ā¬ stamp (Marca Da Bollo)
(last but not least) a document that proves you need to go to another country before receiving your new card. I found this module from the website of Questura that you need to fill - regarding the motivation you can see some examples here. Work trip, school trip and reservation of a plane ticket are some of them. In your case I would try asking the host company to send me a sort of invitation letter that could strengthen the request.